Effects Of Different Fragmentation Conditions On The Fragmentation Region And Time In The Disk Instability Model For Giant Planet Formation

The disk instability model can explain the formation of gas giant planets in protoplanetary disks.According to the disk instability model,the gravitational instability may lead to the fragmentation of protoplanetary disks,which will lead to the formation of clumps controlled by its own gravity in the disk.These clumps then contract to become gas giant planets.The process of planet formation can only occur in the fragmentation region of protoplanetary disks.A critical issue of the disk instability model is whether a disk can fragment or under what condition a disk can fragment.Tang&Jin(2019)points out that the fragmentation conditions used in previous studies are incomplete,and two factors must be considered in fragmentation conditions:effects of the external irradiation on the cooling rate of disks and effects of the isothermal region in disks.They consider effects of the external irradiation and the isothermal region on the fragmentation in disks,and we extend their research.We do a lot of numerical calculations,and we use fragmentation conditions that include different factors to investigate the fragmentation in protoplanetary disks under different fragmentation conditions.The fragmentation conditions we use are as follows:fragmentation criterion without effects of the external irradiation and the isothermal region considered;fragmentation criterion with the isothermal region in disks considered;fragmentation criterion with the external irradiation considered;fragmentation criterion with the isothermal region in disks and the external irradiation considered.We also consider the effect of critical cooling time on fragmentation.We study effects of the isothermal region and the external irradiation on the fragmentation region and time in disks.We use the cooling time criterion with external irradiation and the fragmentation criterion with the external irradiation and the isothermal region to study the effects of the isothermal region on the fragmentation region and time.We find that the inner boundary of the fragmentation region moves inward in the later stage of fragmentation with the effect of the isothermal region considered.We compare the difference of the fragmentation region and time in disks under the cooling time criterion with and without external irradiation.We find that the inner boundary of the fragmentation region in disks with the external irradiation considered moves inward,the amplitude of inward shift is 10 AU.At the same time,the external irradiation can advance the starting time and delay the ending time of the fragmentation in protoplanetary disks.We also study the variation of the inner boundary of the fragmentation region with the three parameters of the molecular cloud cores,namely,mass M_cloud,temperature T_cloudand rotational angular velocity?,under four different fragmentation conditions.For different molecular cloud core properties,we choose the time when the infall of mass from the molecular cloud core ceases.We find that,for the same molecular cloud core properties,the radius of the inner boundary of the fragmentation region without considering the influence of the isothermal region and the external irradiation is larger than those under the other three fragmentation criteria.For different range of molecular cloud core properties,the amplitude of inward shift by the influence of isothermal region and external irradiation are different.For large molecular cloud core properties(T_cloud>13 K;M_cloud>0.5 M_?;?>2.0×10^-14s^-1),the amplitude of inward shift by the external irradiation is larger.We find that the fragmentation region and time increase with the increase of M_cloudand?,and decrease with the increase of T_cloud.However,the characteristics of the fragmentation region and time changing with these three parameters are different:For different molecular cloud core masses,the time evolution of the fragmentation region follows the same evolutionary path in the early stage of fragmentation;for different molecular cloud core temperatures,as the temperature increases,the fragmentation region moves outwards in the early stage of fragmentation;for different?,when?is large,the range of fragmentation region expands,the starting time of fragmentation is advanced and the ending time of fragmentation is delayed.For the same molecular cloud core properties,the starting time of the fragmentation without the external irradiation and the isothermal region considered is later than those in disks under the other three fragmentation conditions.For different molecular cloud core properties,as?increases,the starting time of fragmentation is advanced and the ending time is delayed,which leads to the increase of fragmentation duration;as T_cloudincreases,the starting time and the ending time of fragmentation decrease,but the decrease of the starting time of fragmentation is less than that of the ending time of fragmentation,which leads to the decrease of the duration of fragmentation;as M_cloudincreases,the starting time of fragmentation remains unchanged,but the ending time is delayed,which leads to the increase of fragmentation duration.The variation of fragmentation region and time with molecular cloud properties is related to its influence on the total angular momentum of molecular cloud cores and the rate of the infall from molecular cloud cores.Finally,we also use different critical cooling timescales?to study effects of cooling time criteria with different critical cooling timescales on the fragmentation region and time.We include effects of the external irradiation and the isothermal region in the fragmentation conditions.The numerical results show that the critical cooling timescales that we choose have effects on the inner boundary of the fragmentation region during about 1.2-6.0×10⁵yr,as?increases,the inner boundary of the fragmentation region moves inwards.However,?has no effect on the fragmentation region beyond this time.This indicates that the inner boundary of the fragmentation region is determined by the cooling time criterion after the disk fragments,but as the disk evolves,the inner boundary of the fragmentation region is contained in the isothermal region and is determined by the fragmentation criterion in the isothermal region.